LIBRAR 


UNIVERSITY  OF  CALIFORNIA 


Received 
Accession  No  . 


Clazs  No. 


.  INSTRUCTIONS  FOR  USING 

DOLMAN'S 

...NEW... 

DECIMAL,     SCALE     MEASURE 
PROTRACTOR. 

~"~"~^"lfe«$!S 

(TJHIVSRSITT 

ESSS 

I7M55) 


INSTRUCTIONS  FOR  USING 

DOLMAN'S 

...NEW... 

DECIMAL    SCALE    MEASURE 
PROTRACTOR. 

This  protractor  produces  length  and  position  of  all  lines  and  degrees  of 
angles  required  by  algebra  and  calculus  by  practical  object  lessons  in  scale 
measure,  thereby  solving  millions  of  problems  by  measurement  in  construc- 
tive geometry  without  algebra  or  calculus. 


DOLMAN'S  PROTRACTOR. 


DESCRIPTION  AND   USE    OP    DOLMAN'S    NEW    DECIMAL    SCALE     MEASURE    PRO- 
TRACTOR. 

This  protractor  is  a  semi- circle  with  the  semi-circumference  of  the  pro- 
tractor graduated  90  degrees  from  OM  line  to  the  right,  and  left  to  the 
bottom  line  DD  of  the  protractor.  The  OM,  or  meridian  line,  meets 
bottom  line,  DD,  at  the  middle,  and  in  the  centra  of  the  small  semi-circle 
and  at  right  angles  to  line  DD. 

The  two  arms,  BB,  BB,  must  be  fastened  by  a  needle  at  the  intersection 
of  lines  DD  and  O.  M.  in  the  small  S3mi-circle.  A  parallelogram  two  by 
four  inches,  is  cut  out  of  the  middle  of  the  protractor  parallel  to  sides  LL,  LL 
and  DD,  DD,  and  from  equi-angular  parallels,  and  by  revolving  the  pro  tractor 
one-half  around,  using  the  nesdle  as  a  pivot,  a  four  inch  square,  and  a  circle 
can  be  formed  with  a  common  centre  to  square  and  circle.  The  movable  bar, 
DD,  moves  parallel  to  bottom  line,  DD. 

An  elastic  scale,  seven  inches  long,  is  for  measuring  arcs,  or  any  other 
measurement  of  lines.  The  inside  edges  are  graduated  to  twenty  spaces  to 
one  inch,  and  sidss  LL  LL  read  from  one  at  bottom  line,  DD,  up  to  200  on  out- 
side lines,  and  40  on  inside  lines. 

Bottom  and  top  lines,  DD  DD,  read  right  and  left  from  OM  line  200  on 
outside  lines  and  40  on  inside  lines.  Arms  BB  BB  read  from  center  needle 
outward  350  on  outside  lines  and  70  on  inside  lines.  All  in  edges  are  graduated 
to  twenty  spaces  to  one  inch.  The  graduations  of  degrees  on  the  semi-circle 
are  not  units  of  length.  Their  departure  of  length  of  arc  depends  on  the 
length  of  the  sides  and  number  of  degrees  of  the  angle. 


USE   OF  THE   DECIMAL  SCALE  MEASURE  PROTRACTOR. 

This  protractor  measures  degrees  of  angles  and  gives  the  length  of  every 
line  to  scale  measure,  and  is  a  guide  to  draw  every  line  by,  without  calculat- 
ing the  length  of  any  lines.  No  other  instrument  in  use  at  this  time  gives 
degrees  of  angles,  length  of  lines  and  a  guide  to  draw  the  lines  of  every 
polygon. 

Decimal  is  a  scale  of  which  the  order  of  progression  uniformly  is  ten.  A 
scale  is  a  system  of  measurement  that  small  spaces  are  used  to  represent 
larger  units  of  measure  and  greater  numbers  of  units  proportionally,  viz :  To 
represent  large  area  on  small  space,  as  maps,  charts,  plots,  diagrams,  &c. 

A  decimal  scale  progresses  thus:  If  one-tenth  of  one  inch  represents  one 
unit,  one  inch  in  length  would  represent  ten  units  of  length,  and  one  square 
inch  would  represent  one  hundred  squares  of  one-tenth  of  squares  of  one 
square  inch,  and  ten  square  inches,  or  3  16-100  inches  square  would  represent 
one  thousand  one -tenths  of  inches  in  two  dimensions.  If  we  assume  that 
one-tenth  of  one  inch  of  scale  in  one  dimension  shall  represent  $1,000,  then 
Vne  inch  in  one  dimension  would  represent  $10,000,  and  one  square  inch  in 
two  dimensions  would  represent  $100,000,  and  ten  square  inches,  or  3  16-100 
square  would  represent  81,090,090,  and  ons  hundred  square  inches,  or  ten 
inches  square  would  represent '$10,000,000,  and  one  thousand  square  inches,  or 


DOLMAN'S    PROTRACTOR. 


31  62-100  inches  square  would  represent  8100,000,000,  and  ten  thousand  square 
inches,  or  one  hundred  inches  square,  would  represent  by  scale  measure,  81,- 
030,000,000. 

The  above  explanation  of  decimal  scale  measure  as  applied  to  quantity, 
by  numbers  of  units  compared  with  extension  of  lines  and  angles  are  given  to 
assist  the  mind  to  comprehend  quantity  and  magnitude  as  multiplied  by  10, 
100,  1,000  &c. 

Arithmetic  is  the  science  of  numbers  applied  to  units  of  quantity. 

Geometry  is  the  science  of  measurement. 

Measurement  is,  first,  ascertaining  the  number  of  units  in  a  line  by  com- 
parison in  extension  in  one  dimension  called  distance ;  second,  by  comparing 
square  units  with  area  in  two  dimensions  called  square  measure;  third,  by 
comparing  square  units  with  thickness  extension  in  three .  dimensions  called 
cubic  measure. 

Algebra  is  the  science  of  ascertaining  unknown  numbers  of  units  of  quan- 
tity by  subtracting  one  known  number  of  units  from  other  known  numbers  of 
units,  or  by  adding,  multiplying  or  dividing  or  all  combined.* 

The  following  tables  of  units  are  in  common  use  in  the  United  States,  and 
are  in  the  arithmetic,  viz:  First,  units  of  length,  as  inches,  feet,  miles,  &c. 
Second,  units  of  area,  or  square  units,  as  square  inches,  square  feet,  square 
acres,  &c.  Third,  units  of  volume,  or  cubic  measure,  as  cubic  inches,  cubic 
feet,  &c.  Fourth,  units  of  angles  360  degrees  in  every  circle,  and  each  degree 
may  be  considered  an  angle;  21,600  minutes  in  every  circle,  and  each  minute 
may  constitute  a  separate  angle.  Every  circle  is  divisible  into  (1,296,000)  one 
million  two  hundred  and  ninety- six  thousand  angles  of  one  second  each,  or 
any  number  less  than  the  angular  space  may  be  an  angle.  Fifth,  units  of 
gravity  (weight)  determined  by  comparing  the  volume  of  water  as  a  standard 
of  287-s  cubic  inches  of  water  equals  one  pound  avoirdupois  weight  and  also 
equals  one  pint  of  liquid  measure. 

The  Winchester  Bushel  contains  2150  42-100  cubic  inches,  or  77627-1000 
pounds  of  water  avoirdupois,  or  5760  grains  apothecary  weight. 

One  ounce  of  Troy  equals  480  grains,  and  also  equals  437}^  grains  avoirdu- 
pois weight. 

Sixth,  units  of  duration  of  time  determined  by  motion  of  the  planets,  and 
are  units  of  seconds,  minutes,  hours,  days  and  years. 

Seventh,  units  of  value  are  created  by  law,  and  may  be  cents  and  dollars, 
shillings  and  pence,  or  any  other  unit  desired.  Degrees  of  angles  have  no 
proportion  of  length  as  to  scale  measure  of  departure  of  the  arc  of  any  angle. 
The  right  angled  triangle  is  a  unit  of  comparison  of  degrees  of  angles,  and 
measured  departure  of  angles. 

Every  right  angled  triangle  is  the  one -half  of  a  square,  provided  the  two 
short  sides  of  the  triangle  are  of  equal  length,  and  one  angle  of  the  triangle 
will  contain  90  degrees,  if  the  short  sides  of  the  right  angled  triangle  are  of  dif- 
ferent length,  the  triangle  is  the  one-half  of  an  equiangular  parallelogram. 

A  right  angle  has  two  sides  and  always  contains  90  degrees,  no  more,   no 

*llie  remainder  ttitvr  subtraction  is  the  unknown  <|iiantity. 


DOLMAN'S  PROTRACTOR. 


less. 

Every  right  angled  triangle  has  three  sides  and  three  angles,  and  one  of  the 
angles  is  always  equal  to  90  degrees,  and  the  other  two  angles  are  equal  to  90 
degrees. 

Every  equilateral  triangle  has  three  sides  of  equal  length,  and  three 
angles  of  60  degrees  each. 

Every  scalene  triangle  has  three  sides  of  unequal  length  and  three  angles, 
and  the  three  angles  combined  equal  180  degrees,  and  the  scalene  triangle  can 
always  be  divided  into  two  right  angled  triangles  of  unequal  dimensions. 


Problems  have  but  one  demand — that  is  how  much — and  that  demand  is 
satisfied  by  adding  quantity  to  quantity,  or  by  subtracting  quantity  from 
quantity. 

All  lines  must  have  position,  and  that  position  and  the  relation  to  the 
position  of  other  lines  give  names  to  the  lines  and  cause  the  names  of  lines  to 
change. 

The  triangle,  square  and  hexagon,  are  the  only  regular  polygons  by  which 
the  angular  space  about  a  point  can  be  completely  filled  up. 

Quantity  is  a  general  term  applied  to  every  thing  which  can  be  increased, 
diminished,  measured,  compared,  or  estimated.  It  embraces  numbers  and 
magnitude. 


DEFINITION. 

RADII  IS  THE   PLURAL  OF  RADIUS. 

A  radius  is  any  straight  line  passing  from  the  centre  of  the  circle  to  the 
circumference  of  the  circle. 

Diameter  is  any  straight  line  passing  through  the  centre  of  a  circle  from 
one  side  of  the  circumference  to  the  opposite  side  of  the  circle,  or  any 
polygon. 

Circle,  or  circumference,  is  a  line  which  is  equal  distance  from  a  point 
within,  called  centre. 

Perimeter,  is  any  number  of  straight  lines  that  enclose  a  polygon. 

A  polygon  is  any  diagram  with  three  or  more  sides. 

A  diagonal  line  is  a  straight  line  drawn  from  one  angle  to  an  opposite 
angle. 

A  vertex  is  the  point  where  two  lines  meet,  that  form  angles. 

Base  line,  or  meridian  line,  is  the  beginning  line,  or  line  that  all  other 
lines  conform  to. 

A  perpendicular  line  is  a  line  that  meets  another  line  at  right  angle,  called 
departure. 

A  hypothenuse  line  is  the  longest  line  of  every  right  angled  triangle,  and 
forms  the  third  side  of  the  triangle,  and  in  surveying  land  is  called  the  bearing 
Hne. 


DOLMAN'S  PROTRACTOR. 


Co-tangent,  is  a  line  that  will  meat  the  tangent  at  right  angle  and  starts 
from  the  same  circumference  90  degrees  from  where  the  tangent  starts. 

Co-ordinate  triangle,  means  another  triangle  of  equal  dimensions  of  the 
first  triangle,  and  opposite  to  the  first  trian  416. 

Ratio,  is  that  relation  between  two  quantities  which  is  expressed  by  the 
quotient  of  the  first  divided  by  the  second.  Thus  the  ratio  of  4  to  12  is  12-4, 
or,  12  divided  by  4,  the  ratio  is  3. 

A  proportion  is  an  equality  of  ratios. 

Infinity  in  measurement  is  when  two  lines  approach  so  near  to  each  other 
that  no  perceivable  difierence  can  be  seen. 

Inscribed  means  one  polygon,  or  circle  produced  within  another  circle  or 
p  ;>!ygon. 

Described  means  a  circle  or  polygon  produced  around  another  circle  or 
polygon. 


Applicati-  n  of  the  decimal  scale  measure  protractor  to  the  following  five 
diaijrams,  as  per  instructions,  will  enable  the  student  to  determine  quantities 
and  magnitude  by  measurement  without  algebra,  or  intricate  calculus: 


DIAGRAM  NO.    1. 

This  diagram  represents  first  a  plane,  second  a  sphere,  third  the  propor- 
tion of  area  of  a  circle  to  the  area  of  a  square,  which  is,  circle  area, 7854 

Area  of  square  is,  square .10000 

The  centre  of  this  circle  represents  in  geometry,  first,  the  point  of  hegln- 
ning  of  a  space.     Second,  the  centre  of  a  sphjie  or  the  earth.    The  centre  of 


DOLMAN'S    PROTRACTOR. 


Square  the  two  short  sides,  add  their  products  and  extract  the  square 
root  of  the  product. 

This  earth  is,  in  geometry,  conceived  to  be  the  centre  of  space,  and  all  con- 
ceivable lines  that  start  from  the  centre  of  this  earth  are  considered  in  geom- 
etry as  perpendicular  lines,  and  all  lines  produced  at  right  angles  to  the 
perpendicular  lines  are  considered  as  horizontal  lines.  The  lines  that  are 
conceived  to  pass  from  the  csntre  of  the  earth  to  its  surface  are  represented 
by  a  plummet,  and  the  horizontal  lines  are  represented  by  the  level,  and  are 
also  called  tangants,  if  continued  in  a  straight  line,  and  are  known  as  apparent 
levels,  or  horizontal  lines. 

The  lines  produced  on  the  circumference  of  the  earth  by  the  level  when 
taken  at  different  points  of  the  earth's  surface,  produce  circular  line  s,  called 
true  level,  and  are  always  at  right  angles  to  the  plumb  lines  caused  by  the 
change  of  position  of  the  level  on  the  circumference  of  the  earth  as  the  Iev3l 
will  be  at  right  angle  to  every  plumb  line  at  every  point  of  the  earth's  surface 
and  the  plumb  line  changes  toward  the  centre  of  the  earth  when  moved  from 
one  point  to  any  other  point  on  the  earth's  surface.  Apparent  level,  or  tan- 
gent lines  only  change  four  times,  until  they  meet  and  form  a  square  whoso 
sides  are  equal  in  length  to  the  length  of  the  diameter  of  the  inscribed  circle. 

The  centre  of  diagram  No.  1,  as  a  plane,  represents  the  beginning  point  of 
measurement  by  lines  and  angles. 


TO    ADJUST    THE    PROTRACTOR     FOR    PLOTTING    AND     MEASURING     LINES     AND 

ANGLES. 

Insert  a  needle  or  pin  into  the  holes  in  the  centre  of  the  half  circles  on 
BB,  BB,  with  the  graduated  edges  of  the  arms  right  and  left,  then  insert  the 
needle  into  the  hole  in  the  half  circle  on  bottom  line  DD,  place  movable  bar, 
DD,  inside  the  square  with  the  graduated  edges  parallel  to  bottom  line,  DP, 
then  insert  the  point  of  the  needle  into  the  paper,  and  proceed  to  construct 
lines  and  angles  according  to  requirements  of  the  parts  of  diagram  as  require  d 
by  terms  of  the  problem. 

Meridian,  or  latitude  line,  is  a  line  produced,  or  conceived  north  and 
south.  A  longitude,  or  departure  line,  is  a  line  produced  east  or  west,  and 
departs  from  a  latitude,  or  base  line,  at  right  angle. 

Sine  of  arc,  is  the  number  of  degrees  and  length  of  the  perpendicular 
line  to  a  radius  of  a  circle. 

Co -sine,  is  the  degrees  and  length  of  the  adjoining  side  of  the  triangle 
of  which  the  rxdius  is  a  parallel,  and  the  perpendicular  is  the  sine. 

Secant,  is  the  hypothenuse  of  the  right  angled  triangle  of  the  radius  and 
tangent. 

Co-secant,  is  the  hypothenuss  of  the  complement  angle.* 

Tangent  line,  is  a  line  that  is  produced  at  right  angle  to  a  radius,  and  if 
the  secant  is  at  an  angle  of  45  degrees  to  the  tangent  line,  the  tangent  and 
radius  will  be  of  equal  length.  Otherwise,  the  tangent  will  be  longer  than 

*W hen  the  degrees  of  an  an tfe  are  subtracted  from  90  degrees,  the  rpmaining  degrees  are 
called  the  complement;  and  when  the  degrees  of  an  angle  are  subtracted  from  180  degrees,  the 
remainder  is  called  the  supplement  angle.  See  co-sine,  co-tangent  etc.,  in  diagram  No  4 


DOLMAN'S  PROTRACTOR. 


the  radius.* 

To  trace  the  lines  of  diagram  No.  1,  with  the  protractor  as  an  object  lesson 
in  drawing,  adjust  the  protractor  as  instructed,  then  insert  the  needle  into  the 
centre  of  diagram  No.  1,  place  the  bottom  line,  DD,  of  the  protractor  on  one 
of  the  diameters  of  the  circle  and  movable  bar,  DD,  will  be  parallel  with  the 
tengents  and  chords  of  the  square.  Sides  LL,  LL,  will  be  parallel  at  righ* 
angles  to  lines  DD,  DD,  and  arms  BB,  BB,  will  be  movable  to  coincide  with 
radius  or  bearing  lines.  Movable  bar  DD,  will  indicate  departure,  tangents 
and  chords,  and  sides  LL,  LL,  can  be  moved  to  indicate  latitude  and  BB,  BB, 
will  represent  meridian  lines. 

The  OM  line  will  always  be  at  right  angles  to  DD,  DD,  and  parallel  to  LL, 
LL,  and  to  mark  opposite  parallels  must  not  be  omitted  when  bearings  change 
a  igleS;  and  the  protractor  is  to  be  moved  to  another  bearing.  The  elastic  scale 
will  give  the  length  of  all  lines  and  arcs  to  scale  measure. 

A  thorough  knowledge  of  geometry  can  only  be  acquired  by  producing 
lines  and  angles  to  scale  measure. 

To  construct  a  circle,  use  a  strip  of  card  board  for  a  radius,  use  a  needle 
for  a  centre  pivot,  get  the  length  of  the  radius  from  any  part  of  the  protrac- 
tor, except  the  degrees;  make  a  small  hole  in  the  card  board  for  a  pencil  point 
to  mark  the  circumference  of  the  circle  and  a  needle  for  a  centre. 

To  form  a  square,  use  bottom  line,  DD,  of  the  protractor  cne  side  of  the 
square,  and  one  of  side  LL  for  the  next  side,  then  invert  the  protractor  and 
use  the  same  sides  and  same  length  of  lines  to  construct  the  other  sides  of  the 
square. 

To  construct  parallels,  move  bar  DD  the  distance  from  bottom  line  DD 
that  the  parallels  are  required  apart,  and  see  that  both  ends  of  bar  DD  are  the 
same  distance  from  bottom  line  DD. 

To  construct  a  right  angled  triangle,  place  one  arm  on  the  OM  line  of  the 
protractor,  place  the  other  arm  the  given  number  of  degrees  from  the  first 
arm,  move  parallel  bar  DD  to  the  given  number  on  the  arm  that  represents 
the  given  line,  or  if  the  given  line  is  departure,  move  bar  DD  until  the  given 
number  of  graduations  on  bar  DD  fills  the  space  between  the  two  arms  BB, 
BB  and  the  two  arms  from  bar  DD  to  the  needle  will  be  the  length  of  the  other 
two  sides  of  the  right  angled  triangle. 

When  the  bearing  of  a  right  angled  triangle  is  given,  that  is  the  number 
of  degrees  of  departure  from  the  meridian  line,  and  the  length  of  the  bearing 
line  given  to  obtain  the  length  of  the  latitude  line,  and  the  length  of  the  de- 
parture line  of  the  right  angled  triangle,  place  one  arm  on  the  OM  line  of  the 
protractor,  and  place  the  other  arm  the  number  of  degrees  to  the  right  or  left 
of  the  OM  line,  then  move  bar  DD  parallel  until  the  number  of  graduations 
given  is  found  on  the  arm  that  is  not  on  the  OM  line,  then  the  arm  on  the 
OM  line  between  the  bar  and  needle  will  be  latitude,  and  the  distance  on  bar 
DD  between  the  two  arms  will  be  the  length  of  departure,  and  the  other  arm 
will  give  bearing  distance. 

To  construct  an  equilateral  triangle,  place  one  arm  30  degrees  to  the  right 

*Secant  lines  pass  outside  of  the  circle  and  meet  the  tangent  line. 


BOLIVIAN'S    PROTRACTOR. 


o?  the  OM  line  and  the  other  arm  30  degrees  to  the  left  of  the  OM  line;  mov3 
bar  DD  until  the  distance  on  bar  DD  is  equal  to  the  length  on  each  arm.  The  i 
will  the  three  sides  be  of  equal  length,  and  the  three  angles  contain  60  degree 3 
each.  Move  one  arm  to  OM  and  it  gives  the  perpendicular  of  the  triangle. 

To  construct  a  tangent  square,  or  described  square,  to  a  circle,  construct 
two  diameters  at  right  angles  to  the  circle,  divid  ng  the  circle  into  four  equal 
parts;  place  bottom  line.  DD  and  side  LL  on  the  outside  of  the  circle,  so  that 
the  length  of  the  radius  of  the  circle  on  DD  and  LL  will  meet  two  ends  of 
the  diameters  will  produce  the  first  one -fourth  of  the  square.  Then  move 
the  protractor  to  the  ends  of  the  next  diameters,  and  so  on  until  the  square  is 
completed.* 

A  line  from  the  centre  of  the  circle  to  the  angle  of  the  described  square 
of  a  circle  will  be  a  radius  that  will  double  the  arsa  of  the  first  circle,  and  the 
diagonal  of  the  described  square  will  be  the  length  of  a  square  double  the 
area  of  the  described  square. 


RULES  FOR  CALCULATING  LENGTH    OF    LINES,    NUMBER    OF    SQUARE    UNITS    IN 
AREA  AND   CUBIC   UNITS   OF  VOLUME. 

The  diameter  of  a  circle  given,  required  the  Icngtli  of  the  circumference 
of  the  circle. 

RULE   BY   CALCULATION. 

Multiply  the  length  of  the  diameter  of  the  circle  by  3.1416;  point  off  four 
figures  on  the  right  of  the  product  for  decimals,  and  the  remaining  figures 
will  be  whole  units.  The  circumference  of  a  circle  given,  required  the  diam- 
eter of  the  circle. 

RULE  BY   CALCULATION. 

Add  four  ciphers  to  the  circumference,  then  divide  by  3.1416,  and  point  off 
four  right  hand  figures  for  decimals  in  the  quotient.  The  diameter  of  a  circle 
given,  required  the  area  of  the  circle. 

RULE   BY   CALCULATION. 

Multiply  one-half  of  the  diameter  by  one-half  of  the  circumference,  or 
square  the  diameter  of  the  circle,  and  multiply  that  product  by  the  decimal 
.7854.  Point  off  four  right  hand  f  gures  for  decimals. 

The  two  short  sides  of  a  right  angled  triangle  given,  required  tie  area  of 
the  triangle. 

RULE  BY  CALCULATION. 

Multiply  one  short  side  by  one-half  the  length  of  the  other  short  side. 
The  two  short  sides  of  a  right  angled  triangle  given,  required  the  length  of 
the  bearing  or  hypothenuse. 

RULE   BY   CALCULATION. 

*>ce  Di  'gram  No.  1 


DOLMAN'S  PROTRACTOR. 


Sqaare  the  two  short  sides,  add  their  products  and  extract  the  square 
root  of  the  product. 

RULE  BY  PROTRACTOR. 

Piaca  one  arm  of  the  protractor  on  the  OM  line,  place  the  other  arm  on 
ths  givaa  number  of  degrees  of  departurs  of  the  angle,  move  bar  DD  parallel 
until  the  number  given  for  latitude  on  first  arm,  and  the  other  arm  from 
needle  to  bar  DD  will  be  the  length  of  the  bearing  line. 

Ths  diameter  of  a  circle  given,  required  the  length  of  the  side  of  the 
greatest  inscribed  square. 

RULE  BY  CALCULATION  . 

Multiply  the  diameter  by  the  decimal  .7070,  and  cut  off  four  decimals. 

RULE  BY  PROTRACTOR. 

Construct  a  circle,  construct  two  diameters  at  90  degrees,  dividing  the 
circle  into  four  equal  parts.  The  distance  between  any  two  ends  of  the  diam- 
eters will  be  the  length  of  the  sides  of  the  inscribed  square.  Straight  lines 
drawn  between  the  ends  of  the  diameters  will  construct  the  inscribed  square, 
and  the  four  sides  will  be  four  chords  to  the  four  arcs  made  by  the  two  diam- 
eters of  the  circle. 

The  length  of  the  side  of  a  square  given,  required  the  area  cf  the  square. 

RULE  BY  CALCULATION. 

Multiply  the  length  of  the  side  by  its  own  length.* 

The  area  of  a  square  given  required  the  area  of  a  circle,  whose  diameter 
is  equal  to  the  side  of  the  square. 

RULE  BY  CALCULATION. 

Multiply  the  area  of  the  square  by  the  decimal  .7854,  and  point  off  four 
decimals  on  the  right. 

The  diameter  of  a  circle  given,  required  the  length  of  the  side  of  a  square 
whose  area  equals  the  area  of  the  circle. 

RULE  BY  CALCULATION. 

Multiply  the  diameter  of  the  circle  by  the  decimal  .8862,  and  cut  off  four 
decimals.  • 

The  length  of  the  side  of  an  equilateral  triangle  given,  required  the  per- 
pendicular of  the  triangle. 

RULE  BY  CALCULATION  . 

Multiply  the  length  of  the  side  by  the  decimal  .8660,  and  point  off  four 
decimals. 

RULE   BY  PROTRACTOR. 

Construct  the  triangle  and  measure  the  distance  from  any  one  of  the 
angles  to  the  center  of  the  opposite  side. 


To  multiply  a  number  by  its  own  length  is  called  squaring  a  number. 


10  DOLMAN'S  PROTRACTOR. 

The  ci  -cumf erence  of  a  circle  given,  required  the  length  of  the  side  of  a 
squa  e  equil  in  area  to  the  area  of  the  circle. 

RULE  BY  CALCULATION. 

Add  three  ciphers  to  the  circumference,  then  divide  by  4.442,  and  point 
off  three  decimals  in  the  quotient. 

The  diameter,  or  base,  and  length  of  a  cylinder  given,  required  the  volume 
or  cubic  contents. 

RULE  BY  CALCULATION. 

Multiply  the  square  of  the  base  by  the  decimal  .7854,  and  that  product  by 
the  height  of  the  cylinder,  point  off  four  decimals. 

The  diameter  and  height  of  a  cylinder  given,  required  the  superficial  con- 
tents (area)  of  the  cylinder. 

RULE  BY  CALCULATION. 

Mu  biply  the  diameter  of  the  cylinder  by  3.1416,  and  multiply  that  product 
by  tne  height  of  the  cylinder;  point  off  four  decicals,  and  that  product  will  be 
the  perpendicular  superficial  contents  of  the  cylinder,  less  the  two  ends  of  the 
cylinder;  multiply  the  squares  of  the  diameter  of  the  cylinder  by  two,  and 
multiply  that  product  by  the  decimal  .7854;  point  off  four  decimals. 

The  area  of  a  sphere  is  equal  to  the  area  of  four  circles,  whose  diameters 
are  equal  to  the  diameter  of  the  sphere. 

The  diameter  of  a  sphere  given,  required  the  area  of  the  sphere. 

RULE  BY  CALCULATION. 

Square  the  diameter  of  the  sphere,  multiply  that  product  by  four,  and 
multiply  that  product  by  the  decimal  .7854,  and  point  off  four  decimals. 

The  area  and  diameter  of  a  sphere  given,  required  the  volume,  or  cubic 
contents  of  the  sphere. 

RULE  BY  CALCULATION. 

Multiply  the  area  of  the  sphere  by  one- sixth  of  the  diameter  of  the 
sphere. 

The  perpendicular  of  an  equilateral  triangle  is  three -fourths  the  length  of 
the  diameter  of  its  described  circle,  and  two-thirds  the  distance  from  the 
vertex  to  the  opposite  side  on  the  perpendicular  will  be  the  centre  of  the 
described  circle.  • 

Multiply  the  length  of  the  side  of  an  equilateral  triangle  by  twenty  and 
divide  that  product  by  twenty-three.  This  will  very  nearly  give  the  length 
of  the  perpendicular  of  the  triangle. 

RULE   BY   PROTRACTOR. 

Measure  the  perpendicular  of  the  triangle  with  the  protractor. 
The  length  of  radius  and  degrees  of  arc  given,  required  the  length  of  the 
arc. 

RULE  BY  CALCULATION. 

Multiply  the  radius  by  3.1416,  divide  that  product  by  180,  and  multiply 
that  quotient  by  the  number  of  degrees  of  the  arc.  This  will  givie  the  length 


DOLMAN'S  PROTRACTOR. 


11 


of  the  arc.    Point  off  four  decimals  in  the  last  product.* 

Every  square  is  divisible  into  two  right  angled  triangles  of  equal  dimen- 
sions and  the  two  short  sides  of  each  triangle  will  be  of  equal  length,  and  the 
diagonal  line  that  separates  the  two  right  angled  triangles  will  be  the  hypoth- 
enuse  to  both  right  angled  triangles.  If  one  short  side  of  the  right  angled 
triangle  is  longer  than  the  other  short  side  of  each  right  angled  triangle,  its 
polygon  is  an  equiangular  parallelogram. 

The  side  of  a  sector  whose  angle  is  60  degrees  given,  required  the  length 
of  the  arc.f 

RULE  BY  CALCULATION. 

Multiply  the  length  of  the  side  of  the  sector  by  3.1416;  divide  that  product 
by  three,  and  point  off  four  decimals  in  the  product. 

Co-ordinate  angle  means  another  angle  equal  to  the  first  angle  with 
opposite  bearings. 

The  decimal  scale  measure  protractor  does  not  give  area  and  volume  of 
quantities.  This  protractor  gives  length  of  lines  and  degrees  of  departure  of 
angles,  and  the  form  of  all  angles  and  diagrams. 

All  area  and  volume  are  ascertained  by  multiplying  the  length  of  a  Jin 9  by 
its  own  length,  called  squaring  a  line,  and  multiplying  the  square  product  by 
the  thickness  gives  volume,  or  cubes. 

Multiplying  one  long  side  by  one  short  side  gives  area  of  equiangular  par- 
allelograms 


DIAGRAM  NO.  2. 


Rule  by  Protractor.— Measure  the  arc  with  the  elastic  scale, 
by  Protractor. -Measure  the  arc  with  the  elastic  scale. 


12  DOLMAN'S  PROTRACTOR. 

Diagram  No.  2  is  the  bass  of  the  principle  of  algebra. 

The  length  of  the  radius  of  the  large  circle  of  diagram  No.  2  given, 
required  the  diagram  by  the  protractor,  and  the  area  of  each  separate  part  of 
the  diagram  by  simple  calculation,  viz :  Addition,  subtraction,  multiplication 
and  division. 

RULE  FOR  PROTRACTOR. 

Driw  large  circle,  A,  B,  C,  D,  E,  F,  to  given  radius;  divide  large  circle 
into  six  equal  parts  by  constructing  diameters,  AD,  BE,  FC,  60  degrees  in 
each  division;  place  the  centre  of  the  protractor  at  D  on  large  circle,  and  top 
of  protractor,  OM  line,  on  A;  move  right  arm  20  degrees  to  the  right  from 
OM  line,  and  at  the  point  that  the  arm  crosses  the  second  line,  viz:  BE  win 
be  the  centre  of  first  small  circle;  move  left  arm  20  degrees  to  the  left  of  OM 
line,  and  where  the  left  arm  crosses  the  second  line,  FC,  will  be  the  centre  of 
the  second  small  circle;  move  centre  of  protractor  to  point  B;  move  top  of 
protractor,  OM  line,  to  E ;  move  left  arm  20  degrees  to  left  of  OM  line,  and 
the  point  on  second  line,  DA,  will  be  the  centre  of  third  small  circle.  Move 
right  arm  20  degrees  to  the  right  of  OM  line,  and  the  right  arm  will  meet  the 
centre  of  first  small  circle.  If  the  diagram  is  correctly  constructed,  the  lines 
Y  B,  Y  D,  and  Y  F,  are  the  radii  of  the  three  small  circles,  and  all  the  straight 
lines  of  the  diagram  can  be  measured  by  the  protractor  to  form  the  tangents 
or  described  square  around  the  great  circle.  Place  bottom  line  DD  and  side 
LL  on  the  outside  of  large  circle,  placing  side  DD  at  either  diameter  on  large 
circle  and  move  the  protractor  until  the  length  of  the  radius  of  the  large  circle 
meets  the  circle  that  will  form  the  first  one-fcurlh  of  the  square,  and  the  line 
DD  will  be  first  tangent  line,  and  line  LL  will  be  first  co- tangent  line.  Form 
other  tangents  the  same  way. 


CALCULATION  OP  THE    AREA   OF   THE   SEVERAL  DIVISIONS   OF   PIAGKAM  NO.    2    IY 

ARITHMETIC. 

First  required  is  the  area  of  large  circle. 

RULE. 

Square  the  diameter  of  the  large  circle,  multiplying  that  product  by  the 
decimal  .7854,  anjl  point  off  four  decimals. 

Second,  estimate  area  of  small  circle  by  the  same  rule  which  applies  to 
large  circle. 

Third,  ascertain  area  of  equilateral  triangle  Y,  Y,  Y,  by  multiplying  length 

of  perpendicular  by  one-half  the  length  of  Y,  Y,  Y.     Subtract  one -half  the 

area  of  one  small  circle  from  the  area  of  triangle  Y,  Y,  Y,  and  the  remainder 

equal  the  six  small  divisions  around  the  centre  of  large  circle.    Divide 

A  ^  *?mainder  bv  8ix>  and  Jt  wil1  g^e  the  area  of  one  of  the  small  divisions. 

ie  area  of  the  three  small  circles  and  the  area  of  the  six  small  divisions, 

and  subtract  that  product  from  the  area  of  the  large  circle,  which  will  give 

area  of  the  .six  large  irregular  divisions  of  the  large  circle,  and  divide  that 

•emamder  by  six,  which  will  give  the  area  of  one  of  the  large  irregular  divis- 


DOLMAN'S  PROTRACTOR. 


13 


ions  of  the  circle.  Subtract  the  area  of  the  large  circle  from  the  area  of 
the  described  square.  The  remainder  will  equal  the  four  irregular  divisions 
of  the  square.  That  remainder  divided  by  four  will  give  the  area  of  one  of 
the  four  irregular  divisions  of  the  square. 

Careful  inspection  of  instructions  given  for  diagram  No.  2  will  show  that 
the  twelve  irregular  divisions  of  the  large  circle  can  not  be  measured.  The 
only  process  to  obtain  their  area  is  to  subtract  one  known  area  from  another 
known  area,  either  by  arithmetic  or  by  algebra. 

The  equilateral  triangle,  Y,  Y,  Y,  separates  the  sector,  or  one- sixth  part 
of  the  area  of  each  of  the  three  small  circles;  hence,  three-sixths  equal  one- 
•fcalf  of  the  area  of  one  small  circle,  and  the  remainder  of  the  area  of  triangle 
Y,  Y,  Y,  equals  unknown  quantities  of  the  six  small  divisions,  and  the  six 
large  divisions  and  four  divisions  of  the  square  are  obtained  by  the  same 
rule. 


DIAGRAM  NO.  3. 

Diagram  No.  3  is  given  to  show  the  proportion  and  similarity  of  all  right 
angled  triangles  and  the  use  of  latitude  and  departure  and  the  principle  of 
longitude  sines,  tangents,  etc. 

To  construct  Diagram  No.  3,  place  centre  of  protractor  at  A  and  OM  line 
on  D,  and  bottom  line  DD  will  be  on  A,  B.  Place  right  arm  on  C,  53  degrees 
from  meridian  line  A,  D  ,  N:  A,  C,  will  be  bearing  north,  and  53  degrees  east. 
Let  line  A,  C,  be  50  in  length  to  C.  We  want  to  know  how  far  south  it  is 
from*  C  to  B,  and  how  far  west  it  is  from  B  to  A.  Move  bar  DD  parallel  to  C> 
and  line  A,  D,  will  indicate  30  latitude  north  on  side  LL  and  on  the  arm  placed 

*].alituile  s<unh.  .lf]>anure  \v 


11  DOLMAN'S  PROTRACTOR. 

on  OM  line  on  line  A,  D,  N,  and  bar  DD  will  indicate  40  east  from  D  to  C. 
Move  the  protractor  centre  to  C,  place  line  DD  of  protractor  on  D  of  diagram 
and  OM  line  en  C,  B.  Move  bar  DD  parallel  to  A,  B;  then  will  C,  B  indicate 
south  latitude  from  C  to  B,  and  bar  DD  will  indicate  west  departure  40  from 
B  to  A.  The  solid  line,  A,  C;  C,  B;  B,  A,  may  represent  a  right  angled  trian- 
gle of  land,  or  any  other  quantity  of  area  and  the  length  of  the  lines  may  rep- 
resent feet,  miles,  or  any  other  units  of  measure.  The  dotted  lines  A,  D; 
D,  C;  C,  A,  represent  the  co-ordinate  triangle  A,  D,  C,  and  the  arc  N,  C,  E 
represents  the  one-fourth  of  a  circle  or  90  degrees  from  meridian  line  A,  D, 
N,  to  east  line  A,  B,  E,  and  C,  A  will  be  bearing  south' 53  degrees,  west,  50. 

The  five  divisions  of  equal  distance  in  triangle  A,  C,  B,  are  given  to  show 
the  similarity  of  right  angled  triangles.  Multiply  the  three  sides  of  the  first 
triangle,  10,  8,  6,  by  2,  and  the  triangle  is  increased  to  20,  16,  12.  Multiply  tri- 
angle 10,  8,  6,  by  3,  and  the  triangle  sides  will  be  £0,  24, 18.  The  multiplication 
of  the  length  of  the  three  sides  of  any  right  angled  triangle  does  not  alter  the 
d agrees  of  the  angles;  it  only  increases  the  area  in  proportion  to  the  increase 
of  the  length  of  the  three  sides,  and  to  divide  the  length  of  all  the  sides  of  a 
right  angled  triangle  by  the  same  number,  decreases  area  without  changing 
the  degrees  of  the  angles. 

Hence,  to  divide  the  length  of  any  one  of  the  three  sides  of  a  right  angled 
triangle  by  any  number  that  will  divide  without  a  remainder;  then  find  the 
length  of  the  other  two  sides  of  the  reduced  triangle  by  the  division.  Then 
multiply  the  other  two  sides  of  the  reduced  triangle  by  the  same  number  that 
the  first  side  was  divided  by,  and  it  will  increase  the  right  angled  triangle  to 
its  original  dimension,  and  the  number  used  as  a  d  visor  and  multiplier  will  be 
a  base.  (Logrithms,  have  a  base  of  10,  100,  1,000,  thtt  is  generally  used  as  a 
basis  of  logrithmic  sines,  tangents,  etc.,  wl  i3h  is  a  decimal  basis  of  the  radius 
of  a  circle. ) 

Take  C  as  beginning  of  diagram  No.  3;  thsn  lins  C,  A,  would  read  south 
53  degrees,  west,  50.  By  application  of  the  protractor  to  the  lines,  wculd  show 
that  angle  A,  B,  C  is  37  degrees,  which  would  also  be  found  by  subtracting  53- 
degrees  from  90  degrees.  Thirty-seven  degrees  are  the  complement  of  53  de- 
grees. A,  B.  would  be  departure  east  40,  and  B,  C,  would  be  latitude  north  30 
degr- 

The  area  of  triangle  A,  B,  C,  is  fonnd  by  multiplying  line  A,  B,  by  one- 
half  of  line  B,  C,  or  multiply  one  short  side  of  the  triangle  by  one-half  of  the 
other  short  side.  See  rule. 

The  elastic  scale  .will  ni3asure  ths  length   cf  arc  N,  C,  rrd  r.rc  C,  E.     f.'ee 
rule  for  finding  length  of  arc  by  protractor  and  by  calculation.- 


DOLMAN'S  PROTRACTOR. 


DIAGRAM  NO.  4. 

Diagram  No.  4  differs  from  diagram  No.  3  in  two  particulars,  viz: 

First,  all  the  lines  of  diagram  No.  3  remain  inside  of  their  circle. 

Second,  the  longest  line  of  the  triangle  (bearing  line)  and  degrees  of  an- 
gle ars  given  to  find  the  latitude  and  departure  of  two  short  lines  of  the  right 
angled  triangle. 

In  diagram  No.  4  the  longest  lines  pass  outside  of  the  circle,  and  are  called 
tangent,  co-tangent,  secant,  and  co-secant,  and  the  shortest  sides  never  pass 
outside  of  the  circle,  and  are  given  with  length  and  the  degrees  of  angle  to 
find  the  length  of  the  long  lines.  The  two  short  sides  are  called  sine  and  co- 
sine, and  the  length  of  tangent  and  secant  increase  of  the  number  of  degrees 
of  sine  of  arc  increase,  and  decrease  in  tho  same  way,  when  co-sine  and  tan- 

At  an  angle  of  45  degrees,  sine  and  co-sine  are  of  equal  length;  tangent 
an-1  C3-tang3.it  are.  of  ecju-il  l?:v;lli,  an:l  see.vnt  anil  co-seoant  are  of  equal 
length.' 

At  93  degrees  t-ie  bpunch  of  the  triangle  are  reached  by  either  sine  or  co- 
sin  3/a  li  is  called  infinity'.  See  definition  of  infinity. 


1(5 


DOLMAN'S  PROTRACTOR. 


The  radius  of  the  circle  in  diagram  No.  4  is  taken  as  unity,  and  sine  and 
tangent  form  sides  of  similar  right  angles.  Latitude  in  diagram  No.  3  corres- 
ponds with  sine  in  diagram  No.  4,  and  departure  in  No.  3  corresponds  with 
co-sine  in  No.  4. 

The  length  of  all  lines  of  diagram  No.  4  can  be  constructed  and  measured 
with  the  protractor  by  similar  instructions  as  given  for  constructing  diagram 
No.  3.  Practice  drawing  tangents  and  secants  to  the  radius  of  a  circle  to 
every  five  degrees  of  the  90  degrees  of  the  circle,  and  note  the  rapid  increase 
in  the  length  of  the  tangent  and  secant  when  the  secant  and  tangent  angle 
approaches  90  degrees. 


B 


DIAGRAM  NO.  5. 

Diagram  No.  5  is  given  to  show  how  to  obtain  distance  to  inaccessible 
objects  by  the  right  angled  triangle  as  given  in  co-sines  in  diagram  No.  4  C  is 
first  point  of  observation;  A  is  first  inaccessible  object;  D  is  second  point  of 
observation  to  object  A.  C  is  first  point  of  observation  to  B,  or  second  inac- 
cessible object,  and  E  is  second  point  of  observation  to  object  B.  Required 
the  distance  from  C  to  A  and  the  distance  from  C  to  B  and  the  course  and  dis- 
tance from  A  to  B. 

We  have  a  compass  to  give  angles,*  and  chain  to  give  distance,  C,  D,  the 
compass  says,  course  C,  A,  is  south,  and  the  line  D,  A  is  south,  18  degrees* 

*Mi'nsiin-«l  lim-imi-t  always  be  lakcn  at  ri^ht  ungta  to  the  line  from  observation  to  oi,j«  cf. 


DOLMAN'S  PROTRACTOR.  17 

west,  and  the  chain  gives  measure  eight  to  line  C,  D.  Now  .we  have  a  co-sine 
of  18  degrees  and  eight  measurement  of  sine.  We  now  place  centre  of  the 
protractor  on  A;  place  one  arm 'on  OM  line;  place  the  other  arm  on  18  de- 
grees from  OM  line,  move  bar  DD  parallel  until  eight  spaces  on  bar  DD  fills 
the  space  between  the  two  arms. 

The  arm  on  OM  line  from  needle  or  center  to  bar  DD  will  be  30,  the  dis- 
tance from  C  to  A  in  the  same  unit  of  measure  that  0,  D  was  measured  with 
on  the  ground.  We  find  by  compass  that  the  course  from  C  to  B  is  south,  20 
degrees  west,  and  course  from  E  to  B  is  south,  seven  degrees  west,  giving  a 
co-sine  of  sevejp  degrees  and  measured  line  C,  B  is  10.  Place  centre  of  pro- 
tractor on  B  and  place  one  arm  on  OM  line  of  the  protractor  and  the  other 
arm  seven  degrees  from  OM  line;  move  bar  DD  until  10  graduations  on  bar 
DD  fill  the  space  between  the  two  arms.  The  arm  on  OM  line  from  bar  DD 
to  centre  will  be  32,  the  distance  from  C  to  B;  the  other  arin  will  give  distance 
from  E  to  B. 

To  obtain  course  and  distance  from  A  to  B,  construct  right  angled  trian- 
gles C,  Ji,  A,  and  C,  E,  B;  place  side  LL  of  protractor  on  line  C,  A;  move  pro- 
tractor until  bottom  line .. DD  meets  A;  then  mark  Y.  Change  sides  of  the 
protractor  and  mark  Y  on  the  other  side  of 'line  C,  A,  and  Y,  Y  is  parallel  to 
C,  D.  Place  centre  of  protractor  on  A;  move  bottom  line  DD  of  protractor 
on  Y,  Y;  place  arm  on  B,  and  course  from  A  to  B  will  be  south  73  degrees, 
wast;  and,  distance  from  A  to  B  will  be  twelve  on  the  arm.  Line  D,  C  must 
bs  taken  at  right, angle  to  C,  A,  and  line  C,  E  must  be  constructed  at  right 
a'igle  to  C,  B.  Thus,  we  see  that  having,  the  length  of  one 'line,  and  two  an- 
gles of  any  right  angled  triangle,  or  two  lines  and  one  angle  given,  the  decimal 
scale  measure  protractor  can  give  length  of  th#  other  two  sides  'an'd  angle,  or 
two  angles  and  one  side.  -  -  "^  • 

Tns  dotted  lines  in  diagram  No.  5,  are  given  to  show  co-ordinate  angles 
and  opposite  bearings  of  the  diagram.  To  prove  the  angles  A,  C,  D,  and  B,  C, 
E,  take  same  amount  of  distance  and  area  of  the  circle,  or  opposite  ;dir"e'ctions, 
that  triangles  C,  E,  a,  and  C,  E,  b  take  from  the  circle. 

Any  course  may  be  taken  from  toint:of  observation.  Measured  line  must 
always  be  constructed  at  right  angle  .to) '  object  line;  The  names'  that  line  takes 
in  the  different  diagrams  should  be  remembered,  to  prevent  error  in  calcula- 
tion. 

Note  in  plotting  field  notes .  of  land  opposite  parallels  rbust  be  made  at 
every  angle  that  is  less  QGT  greater  than  90  degrees,  viz:  Fir^t,  to  have  a  par- 
allel mark  to  adjust  the  protractor  at  the  next  angle.  Second,  to  find  latitude 
and  departure  to  the  angle. 

No  survey  is  correct  unless  the  lines  close  by  latitude  'arid  departure,  ex- 
tend as  far  north  as  south,  and  as  far  east  as  west,  called  in  surveying,  north- 
ing and  southing,  and  easting  and  westing. 

This  rule  should  be  well  understood.  The  bearing  of  a  right  angled  trian- 
gle given,  required  the  latitude  and  departure. 

FIRST  RULE   BY   PROTRACTOR. 

Place  one  arm  on  OM  line  and  placa  the  other  arm  on  the  number  of  de- 


IS  DOLMAN'S  PROTRACTOR. 

grees  of  departure  of  the  angle;  move  bar  DD  parallel  to  the  number  on  the 
arm  that  is  not  on  the  OM  line,  and  bar  DD  will  be  departure,  and  the  arm  on 
OM  line  will  be  latitude. 

The  latitude  and  degrees  of  departure  of  a  right  angled  triangle 
given,  required  the  bearing  and  departure  of  the  right  angled  triangle. 

SECOND  RULE  BY  PROTRACTOR. 

Form  right  angled  triangle  on  protractor,  and  the  arm  on  OM  line  will  be 
latitude,  bar  DD  will  be  departure,  and  the  other  arm  will  be  bearing. 

The  departure  of  a  right  angled  triangle  and  degrees  of  departure  of  the 
angle  given,  required  the  latitude  and  bearing  of  the  right  angled  triangle. 

THIRD  RULE  BY  PROTRACTOR. 

Form  triangle  as  before.  The  first  rule  applies  to  line  A,  C,  diagram  No. 
3.  Rule  second  applies  to  perpendicular  in  triangle,  diagram  No.  2,  and  line 
A,  B,  in  diagram  No.  3,  and  tangent  Ar  I  in  diagram  No.  4t  and  lines  Ct  A,  and 
C,  B,  diagram  No.  5. 

Third  rule  applies  to  co-sine  in  diagram  No.  4,  etc.  Line  C,  D  and  C,  E 
in  diagram  No.  5  coincides  with  co-sines  in  diagram  No.  4  and  line  C,  B,  in 
diagram  No.  3. . 

Dolman's  New  Decimal  Scale  Measure  Protractor,  patent  June  10th,  1890, 
produces  length  and  position  of  all  lines  and  degrees  of  angles  required  by 
arithmetic,  algebra  and  calculus  by  practical  object  lessons  scale  measure, 
thereby  solving  millions  of  problems,  by  physical  measurement  in  constructive 
geometry  without  the  assistance  of  algebra  and  intricate  calculus. 

This  protractor  conveys  the  idea  of  numbers  and  magnitude  as  applied  to* 
practical  architecture,  mechanics,  land  surveying,  civil  engineering,  naviga- 
tion, mine  surveying,  irrigation,  hydrography  and  astronomy. 

TLis  protractor  is  a  complete  drafting  outfit  for  tl  e  student,  and  when 
made  of  metal  and  graduated  to  100  to  1  inch  with  vernier  to  read  minutes,  it 
is  the  best  and  most  convenient  practical  protractor  in  use.  The  Decimal 
Scale  Measure  Protractor  gives  double  parallel  Lines  and  when  connected 
form  right  angles. 

All  angles  of  every  polygon  that  are  less  or  greater  than  right  angles  must 
have  a  right  angled  triangle  constructed  or  conceived  to  that  angle  before  the 
area  of  that  polygon  can  be  ascertained,  and  the  area  of  the  constructed  right 
angled  triangle  must  be  ascertained  separate  from  the  area  of  the  other  parts 
of  the  polygon,  and  added  to  complete  the  area  of  the  polygon. 

When  constructing  polygons,  with  the  protractor,  every  angle  of  the 
polygon  that  is  lets  or  greater  than  a  right  angle  must  have  a  latitude  and  de- 
parture line  ascertained  by  leaving  the  bearing  arm  on  the  line  of  the  polygon., 
then  place  the  other  arm  on  the  OM  line  of  the  protractor;  move  bar  DD  par- 
allel until  the  end  of  the  line  is  met  by  bar  DD.  The  distance  between  the 
two  arms  on  DD  will  be  departure,  and  the  distance  from  the  centre  will  be 
the  latitude  and  departure  of  every  right  angled  triangle.  The  latitude  and 
departure  of  every  right  angled  triangle  are  the  two  short  sides  of  the  tri- 
angle. 


DOLMAN'S  PROTRACTOR.  19 

When  the  protractor's  centre  is  moved  to  the  end  of  the  line  to  construct 
another  line  and  angle  to  the  polygon,  the  bottom  line  DD  must  be  placed 
on  the  departure  line  to  preserve  the  parallels  to  the  meridian  or  base  line  o: 
the  polygon,  and  when  the  course  reverses,  the  top  of  the  protractor  must  be 
turned  one-half  around,  so  that  the  parallels  may  not  be  lost  hi  returning  to 
the  beginning  point  of  the  polygon. 

No  polygon  is  completed  until  the  last  line  meets  the  beginning  point 
(called  closing  the  survey  or  polygon.) 

The  latitude  and  departure  lines  of  a  polygon  should  be  indicated  by  dot- 
ted lines,  and  the  length  of  the  latitude  and  departure  lines  should  be  noted, 
that  the  area  of  the  triangle  may  be  computed. 

The  latitude  and  departure  should  be  on  the  outside  of  the  polygon  to 
continue  the  parallels  with  the  protractor.  Co-ordinate  latitude  and  de- 
parture lines  may  be  constructed  on  the  inside  of  the  polygon  to  prevent  con- 
fusion and  error  by  adding  area  to  the  polygon  whose  angles  of  the  polygon 
are  less  than  right  angles. 

The  length  of  either  side  of  a  right  angled  triangle  of  any  conceivable 
length  may  be  reduced  by  dividing  the  length  of  the  side  of  the  triangle  by 
any  number  that  will  divide  it  without  a  remainder,  to  a  number  less  than  the 
graduations  on  the  protractor;  then  find  the  other  two  sides  of  the  triangle 
on  the  protractor,  and  multiply  the  two  sides  thus  found  on  the  protractor  by 
the  same  number  that  the  side  of  the  large  triangle  was  divided  by.  This 
will  give  the  length  of  the  other  two  sides  of  the  large  triangle,  which  is  all 
there  is  in  similar  right  angled  triangles  of  latitude  and  departure,  logrithmic 
sines,  tangents,  etc. 

Dolman's  New  Decimal  Scale  Measure  Protractor  produces  length  and 
position  of  all  lines  and  degrees  of  angles  required  by  arithmetic,  algebra, 
and  calculus  by  physical  lines  and  angles,  solving  and  proving  millions  of 
problems. 

The  question  is  asked  "How  does  the  Decimal  Scale  Measure  Protractor 
solve  and  prove  an  infinite  number  of  problems?"  We  answer  that  the  right 
angled  triangle  is  a  unit  of  comparison  of  measure  between  regular  and  irreg- 
ular polygons. 

All  polygons  are  divisible  into  some  number  of  right  angled  triangles  of 
equal  or  different  dimensions,  and  to  multiply  one  short  side  by  one- half  the 
other  short  side  of  any  right  angled  triangle  gives  the  area  of  the  triangle. 

The  Decimal  Scale  Measure  Protractor  can  be  made  to  give  the  length  of 
all  three  sides  and  the  three  angles  to  every  right  angled  triangle  if  the  length 
of  one  side  and  one  angle  are  given.* 

All  areas  are  determined  either  directly  or  indirectly  by  multiplying  one 
short  side  of  a  right  angled  triangle  by  one-half  of  the  other  short  side.  See 
rule. 

Multiplying  the  length  of  a  line  by  its  length  gives  the  area  of  a  square 
whose  side  equals  the  length  of  the  line,  and  whose  area  is  equal  to  the  two 

*The  right  Jingle  is  always  understood  without  giving  it  when   a  right   angled    triangle    is 
given. 


20  DOLMAN'S  PROTRACTOR. 

right  angled  triangles  of  that  square,  and  the  same  rule  applies  to  equian- 
gular parallelograms. 

The  Decimal  Scale  Measure  Protractor  givei*  double  parallel  lines  to  a, 
meridian  or  base  line,  and  donble  parallel  lines  forrn  right  angles. 

All  angles  of  every  polygon  that  are  less  'or  greater  than  ri  ?ht  angles 
must  have  a  right  angled  triangle  constructed  to1  that  angle  before  the  area  of 
that  polygon  can  be  ascertained,  and  the  area  'of  the  constructed 
right  angled  triangle  must  be  ascertained  separate  from  the  area  of  the  other 
parts  of  the  polygon  and  added  to  complete  the  area  of  the  polygon. 

When  constructing  a  polygon  with  the  protractor,  every  angle  of  the 
polygon  that  is  less  or  greater  than  a  right  ailgle'must  have  a  latitude  and  de- 
parture line  ascertained  by  .-leaving  the  beasring  arm  on  the  line;  then  place, 
the  other  arm  on  the  OM  line  and  move  bar  DD  parallel  until  the  end  of  the 
line  of  the  polygon  is  reached.  Then  the  distance  between  the  two  arms  will 
be  the  departure  and  the  distance  from  the  centre  tb  bar  DD  will  be  the  lati- 
tude of  the  right  angled  triangle.' •'«  The  latitude  and  departure  of  every  right 
angled  triangle  is  the  two  short  sides  of  the  triangle.  When  the  centre  of  the 
protractor  is-  moved  to  the  end  of  the  line^to'coristrtict  another  side  and  angle 
to  the  polygon,  the  bottom  line  DD  must  be  placed  6n  the  departure  line  last 
made  to  preserve  the  parallels  to  ihe 'meridian  or  base  lins,  and  when  the 
course  reverses,  the  top  of  the  protractor  must  be  turned  one -half  around,  so 
that  the  parallels  may  not  bo  lost  in  returning-  to  the  begirining  point  of  the 
polygon  or  survey.  No  survey,  or  polygon  is  correct  unless  the  lines  close  by 
latitude  and  departure.  The  latitude  and  departure  lines  of  a  polygon  should 
be  designated  by  dotted  lines,  and  the  length  of -latitude  and  departure  should 
be  noted,  that  the  area  of  the  triangle  may  be  computed. 

The  versed  sine,  or  perpendicular  line  betweon  arc  and  chord,  is  change- 
able in  length,  viz:  First,  the  length  of  the  versed  sine  is  'always  equal  to 
the  difference  ,  jn  tfre  length  of  the  two  longest  sides  of  the  rig-ht  angled  tri- 
angle. Second*  when  the  length  of  the  sine  is :  added  to  the  length  of  lati- 
tude, their  sum  would  equal  the  radius  of  their  circle.  Third,  when  the  versed 
sine  is  added  to  the  one-half  diagonal  of  an: inscribed  square,  that  line  will 
equal  the  radius  :  of  a  circle  that  will  describe  the  inscribed  square,  and  the 
area  of  the  last :circle  will  be  double  the  area  of  the  first  circle;  and  the  area 
of  inscribed  and  desqribed  circles  can  be  doufcted,  and  the-  area  of  inscribed 
and  described  squares  doubled  ad  infinitum.-  ••  See  diagram  No.  1.  ' 

The  length  of  either  side  of  a  ripfht  angled  triangle  ;bf  any  conceivable 
length  may  be  reduced  by  dividing  toe  leiigiti  of  the '-'sitfe-of  the  triangle  by 
any  number  that  will  clivide  it  with  out-  a.  remainder  to  a  number  less  than  the 
number  of  graduations  on  the  protractor;  then  find  the  other  two  sides  cf  the 
triangle  on  the  protractor  and  multiply  the  two  sides  thus  found'  on  the  pro- 
tractor by  the  same  number  that  the- side  of  the^  larg£  triangle  was  divided  by' 
and  it  will  give  the  length  of  the  other  two  sides  of  the,  large  triangle,  which 
is  all  there  is  in  similar  right  angled  triangles  of  latitude  and  departure,  log- 
rithmic  sines,  tangents,  etc. 

Dolman's  New  Decimal  Scale  Measure  Protractor  produces  length  and 
position  of  all  lines  and  degrees  of  angles  required  by  arithmetic,  algebra  and 


DOLMAN'S  PROTRACTOR. 


21 


calculus  by  physical  lines  and  angles;  solving  arid  proving  millions  of  problems. 
TEXAS   LAND.  MEASURE   TABLE:]        TABLE  OF    THE    GEOGRAPHICAL 

MILES  IN  A  DEGREE  OF  LONGITUDE  AT 
EVERY  DEGREE  OF  LATITUDE  ON  THE 
TERRESTRIAL  SPHEROID,  THE  ELLIPTIC  - 
ITY  BEING  ASSUMED  1-300. 


The  Standard  of  Texas  Land  Meas- 
ure is  the  10  vara  chain  containing  50 
links. 

62tj  inches  equals  1  link. 
1  vara  equals  5  Ijnkg. 
33>a  inches  equal  1  vara. 
237  2-10  varas  equal   UB..mile. 
475  2-10    -<•       •    "     ,  >4     " 
950  4-10      "         '"  :",.^a     " 
1900  8-10 
75  13-100 
1000 

4080  2-10 
3555  5-10 
"2886 
2500 
5000 
1344 

NUMBER  OF 


f'3 

K          ,"fV'» 

r3  '•• 

14  "         :CT     &* 

1  labor, 
i  section. 


t 

1. 

\* 

t 

side 

of  1  acre. 

i 

M 

u  1  labor. 

i 

tt 

"  23  league. 

i  • 

tt  • 

"  U      u 

.- 
t 

it 
it 

'**•  i»       •' 

t 

tt 

n   £      u 

i 

tt 

"  '..section 

DTJ/ 

«2kr   ,•*-, 

•OTJ, 

.  i\  J_J-  ^ 

VARAS    IN 

^. 

c> 


25,000,000  sq.  varas  equal  1  league. 

16,666,66623  sq.  " 

12,500,000      "      " 
18,333^33        <4      Ht 
•6,250,000 
.-l,000i«00        " 

3,613,040       " 

1,806,620  22,-iOO  sq.  va.equail  ^section. 

903726016-100      "      "         u      ^4       u     ' 

451,630  8-100        "      "        "       ^t«  * 

602,173  44-100      "      "         "      1-6     " 

5,645  "      u       '«      1  acre.  - 

4428  697-1000  acres         '  A<      1  league. 

177    •  "  1  labor. 

1111  ^  sq.  in.  equal  1  square  vara. 

7  tq.  It.  &1C'3}3  tq.  in.  'equal  Isq.  vara. 

To  reduce  equatorial  miles  to  statute  miles: 

ItCLE.— Multiply  the  equatorial  miles  by  69  1-8  and  divide  the  produst  by  60. 

To  reduce.var^  to  acres, & 

.  RUL^.--Multiply  the  numbes  of  varas  by  177  J8,  cut  of?  6  decimals  from  the 
product,  the  remaining  figures  of  the  product  will  be  acres  and  the  decimals 
Will  be  fractions  of  an  acre ;  or  divide  number  of  square  varas  by  5,645. 

Measure  all  lines  of  the  diagram  with  the  protractor  to  learn  its  use. 

Every  School  Teacher,  Mechanic  and  Scholar  should  have  one  of  thase 
protractors  with  which  to  practice  drafting. 

The  Wise  County  Protractor  Publishing  Co.,  want  an  agent  in  every  city, 
town  and  county  in  the  U.  S.  to  sell  Dolman's  Protractor. 

Reserved  territory  and  a  liberal  commission  given  to  agents.      Write  for 
special  terms  to  J.  H.  Dolman,  Abilene,  Texas.,  General  Agent. 

The  price  of  the  protractor  is  SI,  postage   free.      Address  all  orders  for 
protractors  to  Roy  B.  Bradley,  Abilene,  Texas. 


Lai 

l°Lorig.|Lat.|l°Long. 

Lat.  l°Long. 

o 

Mile* 

iSJo 

Miles. 

o 

Miles. 

0 

.60*000 

30 

52004 

'40 

30.074 

1 

'59.991' 

9fl 

51.475 

61 

29  162 

.   2 

59.963 

32 

50929 

28241 

3 

59.918 

33 

50  369 

C3 

27311 

4 

59  855 

34 

49793 

64 

26373 

5 

-  r* 

•  59  773 

35 

,  49  202 

25  426 

6 

59.673 

36 

48.596 

<66 

24  472 

•7 

59  5j,6 

37 

47  975 

•67! 

23510 

8 

59421 

^38" 

.47-339 

68 

V2541 

9      59  287 

39 

'46689 

69 

21565 

10. 

,'9095 

I6 

46025 

70 

;  20  581 

11 

58.905* 

<fl 

45846 

71 

"19  ,'92 

12. 

58  698' 

'"42 

44  654 

^2* 

18.596 

13 

58  472 

'43 

43948- 

^TtfP 

17  595 

14 

58229- 

-44 

43*28 

"74 

16  £89 

15 

57.969 

45 

42496 

---?§•••• 

'15.576 

16 

57.690 

46      41,750 

76 

14.560 

17  i    57.H95 

47 

40.991 

77 

13539 

«•  18      57.081 

48 

4«>  220 

78 

12514 

19 

56.751 

49 

39  437 

79 

1!  485 

i 

•20' 

56403 

50    '  38  641 

80      10  452 

:21 

56  038 

51       37.8*4 

81'      9416 

5:2      55.6/)6 

•52      37014 

82        8  378 

23      5.i.258 

53      36184 

83      -7337 

24      54.842 

54      35342 

84 

j6*293 

25      54.411 

55^      84491 

5247 

X6 

§3962 

fttf      33628 

86 

4.199 

27' 

£3497 

57      3.'  754 

,87 

8.149 

28 

53015 

£8      31  870 

88 

-  2  UK) 

29 

52  518 

59.     30  1/77 

89 

•-l.(iBO 

30 

52  U04 

60  ;    30  074 

9U 

•O.COO 

*™iYM. 

HATH  A  WAY'S    IMPROVED    TRAVERSE    TABLE,     WITH     RULES     FOR 
OBTAINING  NATURAL  SINES,    TANGENTS,  ANGLES,  ETC 


Copyright,  1896,  by  C.  F.  HATHAWAY. 


I  olman's  New  Decimal  Scale  Measure  Protractor  is  graduated  to  degrees  and  whole  units  of  lin- 
ear measure.  The  following  rules  and  table  are  given  with  reference  to  the  application  of  the 
Scale  Measure  Protractor  to  diagrams  Nos.  3  a«>d  4  in  instructions  for  using  the  Protractor  in 
constructive  geometry  and  trigonometry.  The  table  shows-  latitude  and  departure  to  lour  deci- 
mal places  for  linear  bearing  1.00  and  for  angular  bearings  from  0  to  90  degrees. 

If  the  angular-bearing  isle  s  than  45  degrees  the  angle  will  be  found  in  the  1st.,  5th:.  rr  9th.  column 
Of  the  table  ai  d  the  lii  e»  r  bt  arinp  »t  the  top  •  r  lotum  <f the<o  i  n  n;  iht  latitude  wil1  be  found 
tn  the  column  headed  lat.  at  the  top  of  the  table,  and  the  departure  in  the  column  headed  dep. 

If  i he  angular  bearing  is  more  than  45  degree*  the  angle  will  be  found  in  the  4th  .8th.  or  12th. 
column  01  the  table  The  latitude  will  be  found  in  the  column  marked  lat.  at  the  bottom,  and  the 
departure  in  the- column  marked  dep  at  the  bottom.  The  latitude,  departure  and  linear  bear- 
ings for  diffV rent  distances  with  the  same  angular  bearings  are  proportional.  1  inear  bearing  s= 
unity,  radius,  secant  and  co  sec-ant.  Latitude  =  na  ural  sines  and  tangents.  Departure  =  co- 
sines and  co-tangents.  Verse  sine  =  the, difference  in  linear  benriiig  and  latitude  Angular 
bearings  =  the  number  ot  degrees  minutes  and  seconds  that  measure  the  angle. 

EXAMPLE     FIRVT. 

Example  1st:  Given  the  angular  beari  ig  11  degrees  and  45  minmes  and  linear  bearing  31. 
required  the  latitude  and  departure  of  the  triangK  Rule  1st. :  In  the  table  opposite  11  degrees  45 
minutes  we  find  the  decimal  .9790  in  thj  latitud  >  column  and  decimal  2  K56  in  the  departure  col  - 
umn.  Multiply  .9790  by  bearing  31  =  30.3490;  and  2036  by  31  =6. 3116  required  latitude  and  depar- 
ture. 

EXAMPLE    SECOND 

Example  2nd:  Required  the  latitude  and  linear  bearing  to  g;ven  linear  departure  26  and 
the  angular  bearing  19  degrees  and  15  minutes.  Rule  2nd:  In  the  table  oppos.te  to  19  degree* 
and  15  minutes  we  find  decimal  .9441  in  the  latitude  column  and  .3297  in  the  departure  column 
Divide  the  i  iven  departure  25  by  .3297  =75.8265  the  required  bearing.  Multiply  7R  8265  by  .9441 
=  71  5877  latitude  required. 

EXAMPLE    THIRD 

Example  3rd:  Latitude  16  34  and>angular  bearing  37  degrees  and  30  minutes  given:  required 
linear  bearing  and  departure.  Rule  3r  1:  iu  the  tubl-i  op.>o-site  to  17  d  >g  e  -s  a  id  30  minutes  in 
latitude  column  we  find  decimal  .7984  and  .6088  in  departure  column.  Divide  the  given  latitude 
16.34  by  .7934  =  20  5949  the  required  bearing,  and  multiply  the  liHear  bearin^O  5949  1)^.6088  = 
12. 5381  the  required  departure.  t) -40U-1 W  ^  />•/> 

EXVMPLE    FOURTH.    K  > 

Example  4tn  :  Given  linear  bearing  600  and  linear  departure  100:  Required  the  angular 
bearing.  Rule  4th:  divide  the  given  departure  IOJ  by  given 4MMMM  600  =.1666.  In  the  table  we 
find  the  nearest  number  to  the  quotient  to  be  the  decimal  .1650  opposite  to  »  degrees  and  80 
minutes,  and  .1693  opposite  to 9  degrees  and  4$  minutes,  subtracting  the  quantities  We  have  15 
minutes  equals  0043;  divide  .0013  by  IS  minutes  equals  .00028666  the  tabular  dffe  re  nee  for  1  miu- 
ute.  Subtract  1650  from  1666  equals  0016  divided  by  .00028666  equals  5.581*  minutes.  .5815 
multiplied  by  60  equals  34.89  seconds.  Combining  the  quantities  we  have  9  degrees,  3/.  minutes 
and  34.89  seconds  for  the  required  angular  bearing. 

EXAMPLE  .  FIFIH. 

Example  5th:  First  operation:  Required  the  verse  sine,  (the  verse  sine  i«  the  difference  be- 
tween the  latitude  am  linear  bearing)  for  linear  bearing  31  and  angular  bearing  11  degrees  and  45 
minutes.  By  ruie  I  we  find  the  latitude  to  be  30. 3490  subtract,  and  the  difference  .6510  is  the 
ve  se  sine:  Second  operation:  required  co-tangent  to  the  same  linear  and  angular  bearing 
The  given  bearing  31  becomes  latitude  and  is  worked  by  rule  3rd.  The  co-tangent  is  6  4469  and 
theco-se<*ntxi.6650;  third  operation:  required  the  natural  tangent  and  secant  to  the  same  linear 
and  angular  tearing.  Substitute  the  linear  bearing  31  for  latitude  to  natural  tangent  and 
secant.  The  angular  bearing  of  the  tangent  and  secant  is  found  by  subtracting' the.  given  angle 
11  degrees  and  45  minutes  from  90  degrees  equals  78  degree*  and  15  minutes,  therefore  read  the 
columns  from  the  bottom  and  proceed  as  in  rule  3rd.  tangent  is  149.0618,  and  the  secant  152.25931 
The  latitude  and  departure  given  to  find  linear  bearing:  rule  square  the  latitude  and  departure 
add  them  and  extract  the  square  root. 


Course' 

Dist.  1. 

Course.!  1  P*8* 
-o-HILat. 

;.  1. 

Course.j 

Dist.  1. 

Lat.  |  Dep. 

Dep. 

Lat.  Dep.j 

O  1 

0   1 

0  15 

1.00000.0044 

45 

15  ;  {079648  0.2630 

45 

id 

0.86380.5038; 

45 

30 

0000  0087 

30 

30  1  9636 

2672 

30 

30 

8616  5075 

30 

45 

0.9999  0131 

15 

45  !  9625 

2714 

15 

45  i  8594  5113 

15 

1  0 

9998  0175 

!  89  0 

16  0   9613 

2756 

74  0 

31  o!  8572  5150 

59  0 

15 

9998  0218 

45 

15 

9600 

2798 

45 

id 

8549  5188 

45 

30 

9997;  0262 

30 

30! 

9588 

2840 

30 

30 

8526  5225 

30 

45 

9995  0305 

15 

45 

9576 

2882 

15 

45 

8504  5262 

15 

2  0 

9994  0349 

88  0 

17  0 

9563 

2924 

73  0 

32  0 

8480  5299 

58  0 

15 

9992  0393 

45 

15 

9550 

2965 

45 

15 

8457  5336 

45 

30 

9990  0436 

30 

30 

9537 

3007 

30 

30j 

8434  5373 

30 

45 

0.99880.0480 

15 

45 

0.95240.3049 

15 

45!  0.8410  0.5410    15 

3  0 

9986  0523 

j  87  0 

18  0 

9511  3090t 

i  72  0 

33  0'  8387'  5446 

57  0 

15 

9984  0567 

45 

15 

9497 

3132 

46 

15M  8363  5483 

45 

30 

9981;  0610 

30 

30 

9483 

3173 

30 

30!  1  8339  5519 

30 

45 

9979  0654 

15 

45 

9469 

3214 

15 

45 

8315  5556 

15 

4  0 

9976  0698 

86  0 

19  0 

9456 

3256 

71  0 

34  0!  8290  5592   56  0 

15 

9973  0741 

45 

15 

9441  3297 

45 

15' 

8266  5628    45 

30 

9969  0785 

30 

30   9426  3338 

30 

30 

8241  5664    30 

45 

9966  0828'    15 

45! 

9412 

3379 

15 

45!!  8216  5700    15 

5  0 

9962  087211  85  0 

20  0 

9397 

3420 

70  o 

35  ti 

8192  5736   55  0 

15 

0.99580.0915    45 

15 

0.9382  0.3461 

45 

15j  0.8166  0-5771    45 

30 

9954  0958    30 

30 

9367 

3502 

30 

30 

8141  5807    30 

45 

9950  1002J  i    15 

46 

9351 

3543 

15 

46;  i  8116  5842    15 

6  0 

9945  1045 

I  84  0 

21  0 

9336 

3584 

!  69  o 

36  o'l  8090  5878 

54  0 

15 

9941  1089 

45 

15 

9320 

3624 

45 

15  S064  5913 

45 

30 

9936  1132 

30 

30 

9304 

3665 

30 

30 

8039  5948    30 

45 

9931  1175 

15 

45 

9288 

3706 

15 

45 

8013  5933    15 

7  0 

9925  1219   83  0 

22  0 

9272 

3746 

68  o 

37  0 

7986  6018  '  53  0 

15 

9920  1262    45 

15 

9255 

3786 

45 

15 

7960  6053    45 

30 

9914  1305    30 

30 

9239i  3827 

30 

30 

7934  6038    30 

45 

0.99090.1349!    15 

45 

0.9222  0.3867 

15 

45  0  7907  0.6122    15 

8  0 

9903  1392 

i  82  0 

23  0 

9205 

3907 

67  0 

38  0 

7880  6157   52  0 

15 

9897  1435- 

45 

16 

9188  3947 

45 

15 

7853  6191    45 

30 

9890  1478 

30 

30 

9171 

3987 

30 

30 

7826  6225 

30 

45 

9884  1521 

15 

45 

9153 

4027 

15 

45 

7799  6259 

15 

9  0 

9877  1564 

i  81  0 

24  0 

9135  4067 

66  0 

39  0 

7771  6293   51  0 

15 

9870  1607 

45 

15   9118  4107 

45 

15 

7744  6327 

45 

30 

9863  1660 

30 

30   9100 

4147 

30 

30 

7716  6361 

30 

45 

9856  1693 

15 

45   9081 

4187i 

15 

45 

7688  6394 

15 

10  0 

9848  1736 

80  0 

25  0!  9063 

4226 

i  65  0 

40  0 

7660  6423 

50  0 

15 

0.98400.1779 

45 

15  0.90450.4266 

45 

15 

0.76320.6451 

45 

30 

9833  1822 

30 

30 

9026  4305 

30 

30 

7604  6494 

30 

46 

9825  1865 

15 

45 

9007  4344 

15 

45 

7576  6528 

15 

11  0 

9816  1908 

i  79  0 

26  0 

8988 

4384 

i  64  0 

41  0 

7547  6581 

49  0 

15 

9808  1951 

46 

15 

8969 

4423; 

45 

15 

7518  6593    45 

30 

9799  1994  i    30 

30 

8949 

4462 

30 

30 

7490  6626    30 

45 

9790  2036    15 

46 

8930 

4501 

15 

45 

7461  6659    15 

12  0 

9781  2079  78  0 

27  0 

8910 

4540 

1  63  0 

42  0 

7431  6891   43  0 

15 

9772  2122]!    45 

15 

8890 

4579 

45 

15 

7402  6724    45 

30 

9763  2164    30 

30 

8870 

4617; 

30 

30 

7373  6756    30 

45 

0.97530.22071!   15 

45 

0.8850  0.4656 

15 

45  0.7343  0.6788    15! 

13  0 

9744  2250|!  77  0 

28  0 

8829 

4695 

!  62  0 

43  0 

7314  6820   47  0 

15 

9734:  2292    46 

15 

8809 

473S 

45 

15  7284  6852'    45 

30 

9724  2334!  !   30 

30 

8788 

4772. 

30 

30 

7254  6884    30 

45 

9713  2377]  !   15 

46 

8767  4810 

15 

45 

7224  6915    15 

14  0 

9703,  24191  76  0 

29  0 

8746 

4848 

61  0 

44  0 

7193  6947   46  0 

15 

9692  2462!    45 

15 

8725  4886 

45 

15 

7163  6978    45 

30 

9681  2504    30 

30 

8704 

4924 

30 

30 

7133  7009    30 

45 

9670  25461    15 

45 

8682 

4962 

15 

45 

7102  7040    15  i 

15  0 

9659  2688 

i  75  0 

30  0 

8660 

6000 

60  0 

450 

7071  7071   45  0; 

Dep.  |  Lat. 

pL_L 

Dep. 

|  Lat. 

0   1 

Dep.  Lat. 

Dist.  1. 

jCourge. 

Dist.  1. 

Course. 

1 

Dist.  1.  i  ..Course." 

HATHA WAY'S    IMPROVED    TRAVERSE    TABLE,     WITH     RULES     FOR 
OBTAINING  NATURAL  SINES,  TANGENTS,  ANGLES,  ETC 

Copyright,  1296,  by  C.  F.  HATHAWAY, 


I  olmau's  Now  Decimal  Scale  Measure  Protractor  is  graduated  to  degrees  and  whole  units  of  lin- 
ear measure.    The  following  rules  an  datable  are  given  ^:it-h   reference  to  the  application  of    the 
Scale  Measure  Protractor  to  diagrams  No's.  3  and  4  in  instructions,  for   using    the   Protractor  in 
constructive  geometry  and  trigonometry.     The  table  shows  latitude  and  departure  to  lour  deei- 
.  mal  places  for  linear  bearing  1.00  and  for'angular  bearings  from  6  to  90  degrees. 

Ii  <he  angular  bearing  isle  s  than  45  degrees  the  angle  will  be  found  in  the  1st.",  5th.,rr9th.  column 
Of  the  table  ai  d  the  lii  e:  r  bia$^r*»1  the  lop  <  r  botu  m  r  f  the  cohmn ;  the  latitude  wil1  be  tound 
in  the  column  hea.:l<d  !at.  at  the'top  of  the  table,  and  the  departure  in  the  column  headed  dep. 
If  'he  angular  bearing  is  more  than  45  degrees  the  angle  will  be  found  in  the  4th.,8th.  or  12th. 
column  oi  the  table  The  latitude  will  be  found  in  the  column  marked  lat.  at  the  bottom,  and  the 
departure  in  the  column  marked  dep.  at  the  bottom.  The  latitude*  Departure  and  linear  bear- 
ings for  different  distances  with  the  same  angular  bearings  are  proportional.  linear  bearing  — 
unity,  radius/sceant  and  co  secytfjit.  Latitude.  ==  natural  sines  and  tangents.  Departure  =  co- 
sines aud  co-tangents.  Verse  sfn:e  =  the  difference  in  linear  bearing  and  latitude  Angular 
bearings  =  tire  number  oi  degrees  minutes  and  seconds  that  measure  the  angle. 

:      EXAMPLE     FIR>T. 

Example  1st:  Given  t'he  Angular  beari.-^gll  degrees  and  45  minutes  and  linear  bearing  31; 
requirf  d  the  latitude  and  departure  of  •th#:triangU-  Rule  1st. ;  In  the  table  opposite  11  degrees  45 
minutes  we  find  the  decimal  .9790  in  tlyj  latitud  ; -column  and  decimal  2036  in  the  departure  col- 
umn. Multiply  ,9790  by  bearing  31  =  30:3490;  aud  2036  by  31  =  6.S116  required  latitude  and  depar- 
ture. '-..'••  •"  -•  i  •- 

EXAMPLE  ".SECCpS'p- 

Example  2nd:  Required  the  latitude  and  linear,  bearing  to  given  linear  departure  25  and 
the  angular  bearing  19  de'gree*  and  15  minutes.  Rule  2nd:  In  th^ table  opposite  to  19  degrees 
and  15  minutes  we  find -decimal  .9441  in  the  latitude  column  and'!.,3297  in  the  departure  column 
Divide  the  i  iven  departure  25  by  .3297  =75.8265  the  required  bearing.  Multiply  75.8266  by  .9441 
.=?  71  5877  latitude  required. 

EXAMPLE    THIRD 

Example  3rd:  Latitude  16  3  4  and  angu.laV  bearing  37.  degrees  and  30  minutes  given;  required 
linear  bearing  and  departure.  '  Rttle3r-l:  in  tli;  bible  opposite  to  37  d  ;g  :e;-s,  a .'id  30  minutes  in 
.latitude  column  we  fiiil decimal .7934  and  .6088  in  departure  column.  Divide  the  given  latitude 
16.34  by  .7934  =  20.5919  the^required  bearing,  and  multiply  the  linear  bearing  20.5^49  by  .6088  = 
:  12 .53S1  the  required  <1<  parture.  ,,•  &  J^A/^tWA  (&UV 

EX  vMPLE    FOURTH.  JJ  "f  * 

"  Example  4th  :  Given  linear  bearing  t>0j£ •••.- ojid  linear  -departure  100:  required  the  angular 
bearing.  Rule  4th:  divide  the  given  departure  IOQ  by  given  JMMlftf  600  =.1666.  lathe  table  we 
find  the  nearest  number  to  the  quotient  to  be  the  decimal  .1650  opposite  to  9  degrees  and  30 
minutes,  and  .1693  opposite  to 9  degrees  and  45  minutes,' subtracting  the:  quantities  we  have  15 
minutes  equals  .0043;  divide  .0048  by  15  minutes  equals  .00028666  the  tabular  difference  for  1  min- 
ute. Subtract  .1650  from  1666  .equals  0016  divided  by  .00028666  equals  5,5815  minutes.  .5815 
multiplied  by  60  equals  34.89  seconds.  Combining  'the  quantities  we  have  9  degrees,  35  minutes 
and  34.89  seconds  for  the  required  angular  bearing. 

.        -  EXAMPLE     FIFTH. 

Example  5th:  First  operation:  Required  the  verse  sine,  (the  verse  sine  is  the  difference  be- 
tween the  latitude  am  linear  bearing)  for  linear  bearing  31  and  angular  bearing  11  degrees  and  45 
minutes.  By  rule  1  we  find  the.  latitude,. to  be.30.3490.  subtract,  nnd  the  difference  .6510  is  the 
ve-^esine:  Second  operation:  required  co-tangent  to  the  same  linear  and  angular  bearing 
The  given  bearing  31  becomes  latitude  .and  is  Avorked  by  rule  3rd.  The  co-tangent  is  6.4469  and 
the  co-secant  Hl.6650;  third  operation :  required  the  natural  tangent  and  secant  to  the  same  linear 
and  angular  bearing.  Substitute  the  linear  bearing  31  for  latitude  to.  natural  tangent  and 
secant.  The  angular  bearing  of  the  tangent  and  secant  is  found  by  subtracting  the  given  angle 
11  degrees  and  45  minutes  from  90  degrees  equals  78  degrees  and  15  minutes,  therefore  read  the 
columns  from  the  bottom  and  proceed  as  in  rule  3rd.  tangent  is  149.0618,  ;aud  the  secant  152.2593. 
The  latitude  and  departure  given  to  find  linear  bearing:  rule  square  the?latitude  and  departure 
add  them  and  extract  the  square  root. 


TTHI71 


Photomount 

Pamphlet 

Binder 

Gaylord  Bros.,  Ir 

Makers 

Stockton,  Call 
PM.  JAN-  21.  1»» 


Dolman, 


Instruqtior  s  for  us  ing 
Dolman  s  n<».7-  .  -prQ-hr«/v 


NOV  b     lij 
1936 


63970 


QC100 
.5 

J)6 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


